We summarize the definition of bulk winding numbers for the second parent model introduced in Sec. .
Following Ref. [1]}, we first transform the Floquet operator \({\cal U}\)  into two symmetric
time frames \(a\)  and \(b\)  by shifting the initial time of evolution from \(t=0\)  to \(t=1/4\)  and \(t=3/4\) .
The Floquet operators in these time frames take the forms \({\cal U}_a=e^{-i{\cal H}_2/4}e^{-i{\cal H}_1/2}e^{-i{\cal H}_2/4}\) 
and \({\cal U}_b=e^{-i{\cal H}_1/4}e^{-i{\cal H}_2/2}e^{-i{\cal H}_1/4}\) .
Performing Fourier transforms from position to momentum representations, we obtain
\({\cal U}_\alpha =\sum _{k_x,k_y}|k_x,k_y\rangle {U}_{\alpha }(k_x,k_y)\langle k_x,k_y|\)  with \(\alpha =a,b\) .
In the tensor product form, we have \({U}_\alpha (k_x,k_y)={\cal U}_0(k_x)\otimes {\cal U}_\alpha (k_y)\) , where
\({\cal U}_0(k_x)=e^{-i{\cal H}_x(k_x)}\) , \({\cal U}_a(k_y)=e^{-i{\cal H}_{y2}(k_y)/4}e^{-i{\cal H}_{y1}(k_y)/2}e^{-i{\cal H}_{y2}(k_y)/4}\) ,
and \({\cal U}_b(k_y)=e^{-i{\cal H}_{y1}(k_y)/4}e^{-i{\cal H}_{y2}(k_y)/2}e^{-i{\cal H}_{y1}(k_y)/4}\) .
The \({\cal H}_x(k_x)\) , \({\cal H}_{y1}(k_y)\)  and \({\cal H}_{y2}(k_y)\)  are Fourier transforms of the
Eqs. (REF )–(REF ) in the main text. \({U}_{\alpha }(k_x,k_y)\)  has the CS
in the sense that \(\Gamma {U}_{\alpha }(k_x,k_y)\Gamma ={U}_{\alpha }^{-1}(k_x,k_y)\)  for \(\alpha =a,b\) ,
where \(\Gamma =\sigma _z\otimes \sigma _y\) . In our model, \({\cal U}_0\)  simply describes the evolution operator of
a Su-Schrieffer-Heeger model in its topological flat band limit, which possesses a winding number \(w=1\) .
Taking the Taylor expansion of \({\cal U}_\alpha (k_y)\)  yields
\({\cal U}_\alpha (k_y)=\cos ({\cal E})-i(d_{\alpha x}\sigma _x+d_{\alpha z}\sigma _z)\) ,
for which another winding number can be defined as
\(w_\alpha =\int ^{\pi }_{-\pi }\frac{dk_y}{2\pi }\frac{d_{\alpha x}\partial _{k_y}d_{\alpha z}-d_{\alpha z}\partial _{k_y}d_{\alpha x}}{d_{\alpha x}^2+d_{\alpha z}^2}\)  for \(\alpha =a,b\) .
Put together, we obtain a pair of winding numbers \((\nu _a,\nu _b)=w(w_a,w_b)\)  for the Floquet operators \(({\cal U}_a,{\cal U}_b)\) .
Their combination results in the integer topological invariants \(\nu _0=(\nu _a+\nu _b)/2\)  and \(\nu _\pi =(\nu _a-\nu _b)/2\)  of the
two-dimensional parent system \({\cal U}\) , which are related to the numbers of Floquet corner modes at zero and \(\pi \) 
quasienergies \(n_0\)  and \(n_\pi \)  through the relations \((n_0,n_\pi )=4(|\nu _0|,|\nu _\pi |)\)  [1]}.
Following the analysis in Sec. , \((\nu _0,\nu _\pi )\)  could also take count of the numbers of
fractional-quasienergy corner modes in the \(q\) th-root descendants of the parent model \({\cal U}\) 
so long as the chiral symmetry \(\Gamma \)  is preserved.
